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Capstone Projects Involving Computational Physics

Capstone Projects Involving Computational Physics. J effrey W. Emmert February 20, 2017. Experiment. Physics Research:. Theory. Design, build, observe. Hypothesize, model, predict. Computation. Visualize, simulate. Computation….

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Capstone Projects Involving Computational Physics

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  1. Capstone Projects Involving Computational Physics Jeffrey W. Emmert February 20, 2017

  2. Experiment Physics Research: Theory • Design, build, observe • Hypothesize, model, predict Computation • Visualize, simulate

  3. Computation… • Can be used to investigate models and scenarios that are too difficult or time consuming to evaluate by hand • Offers an alternative to real experiments that are too expensive or hazardous to perform “The purpose of computing is insight, not numbers.” - R. W. Hamming

  4. Why a Computation Capstone Project? • Generally inexpensive, requiring few resources • Can be relatively simple and accessible for undergraduate students • Develops programming skills, which are often useful when students seek employment

  5. Partnership for Integration of Computation into Undergraduate Physics (PICUP) • Seeks to expand the role of computation in the undergraduate physics curriculum • Provides exercise sets that can be downloaded and modified • Offers week-long faculty development workshops • Visit http://www.compadre.org/PICUP/

  6. Recent undergraduate computational physics projects at Salisbury University: • Computer Modeling of Adsorbate Configurations, Lisa Dean and Christian Schwarz • Dynamics of a Double Pendulum with Isosceles Triangle Components, David Binkowski and Zachary Jackson • Packing Disks Optimally, Sarah Confrancisco • Effect of Starting Location on Clusters Formed by Diffusion-Limited Aggregation • Fractals Hidden in the Dynamics of the Compound Double Pendulum , Louise Coltharp , May Palace Faculty mentors include Dr. Jeffrey Emmert and Dr. Gail Welsh

  7. Diffusion-Limited Aggregation • DLA is a growth process • Initialized by a seed particle • Cluster forms as additional particles undergo random walks and attach themselves, one-by-one, to the cluster • Random walk step size affects characteristics of the resulting cluster • We chose to create an off-lattice, two-dimensional simulation of DLA using identical particles

  8. Starting Location: Common Model • Each particles diffuses from a “birth” circle • The birth circle is kept far from the growing cluster

  9. Starting Location: Random Start • Each particle diffuses from a different random “birth” location near the growing cluster • The birth location may not overlap the existing cluster

  10. Characterizing the Clusters • Radius of gyration () • Measure of overall cluster compactness • Can be used to find the cluster’s fractal dimension • Lacunarity () • Measure of how uniformly the particles are distributed • A circular box is centered on each particle and the number of neighboring particle centers enclosed are counted

  11. Common Model 104 particles per cluster step size: 2 step size: 8 step size: 32 Random Start

  12. Common Model Random Start 104 particles per cluster

  13. Radius of Gyration vs. Step Size (106particles per cluster) Average RoG (particle diameters) Random Walk Step Size (particle diameters)

  14. Lacunarity vs. Step Size (106particles per cluster) Average Lacunarity Random Walk Step Size (particle diameters)

  15. The Compound Double Pendulum • Simple mechanical system • Exhibits rich dynamical behavior that is deterministicallychaotic • Future behavior is fully determined by initial conditions • Small differences in initial conditions yield widely diverging outcomes • Differential equations of motion can be derived via Lagrangian mechanics • Time evolution found by numerically integrating the equations of motion

  16. Plotting a “Flip Portrait” double-rod pendulum with and • Initialize the system at rest for an array of and values • For each initial state, evolve in time until a flip occurs (when one arm inverts) • Color the point associated with the initial state according to how long it took until the flip occurred • White areas: initial conditions for which no flips are observed • White, lens-shaped region: the “forbidden zone,” where flip events are energetically impossible

  17. Primary vs. Secondary Flips double-rod pendulum with and

  18. Three Cases of Pendulum Parameters Secondary arm first-flip portraits: double-rod pendula with

  19. References: • R. W. Hamming, Numerical Methods for Scientists and Engineers (McGraw-Hill Book Company, 1962). • J. S. Heyl, The Double Pendulum Fractal, August 2008 (unpublished). • P. Meakin, Fractals, Scaling, and Growth Far from Equilibrium (Cambridge University Press, 1998). • E. P. Rodrigues, M. S. Barbosa, and L. da F. Costa, Phys. Rev. E 72, 16707 (2005). • S. Strogatz, Nonlinear Dynamics and Chaos (Perseus Books Publishing, 1994). • T. A. Witten, Jr and L. M. Sander, Phys. Rev. Lett. 47, 1400 (1981).

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